E.g. u = (2, 3) and v = (4, 1). The weighted Euclidian inner product can be given as 〈,〉 = 11 + 222.
Naturally $$(u.v) = 1(2\times4) + 2(3\times1)$$
$$(u.v) = 8 + 6$$ $$(u.v) = 14$$
But what about the projection of u onto v?
Do we multiply everything by the weighted inner product? Or just the (u.v) part and the (v)^2 part? (i.e. everything except the single v component)
I think it is the latter. i.e. $$PROJvU = \frac{u.v}{v^2}v$$ $$PROJvU = \frac{14}{34}(4,1)$$ $$PROJvU = \frac{7}{17}(4,1)$$
(Alternative answer for multiplying everything would be $$\frac{7}{17}(4,2))$$
Please let me know if my instincts are correct. Thank you